1. Field of the Invention
The present invention relates to a surface shape measuring apparatus used to measure the shape of a curved surface for an aspherical lens or the like.
2. Description of Related Art
The surface shape measuring apparatus used to measure the shape of a curved surface for a lens or the like is configured to measure the radius of curvature, in the case where the surface shape of a measurement object is approximated with a two dimensional shape, whether in tracer type or non-tracer type.
For example, by tracing the surface of the measurement object with a probe (two dimensional scanning), the two dimensional data (x, z) around an origin in the two dimensional orthogonal coordinates as predefined is measured. Then, by fitting the obtained data to a general expression (1) of a circle by the method of least square, the coefficients g, h and c defining the circle are calculated. On the basis of the obtained coefficients g, h and c, the center coordinates (xe2x88x92g, xe2x88x92h) of the circle are calculated, and the radius of curvature is calculated in accordance with an expression (2).
xe2x80x83xe2x88x92c=x2+z2+2gx+2hzxe2x80x83xe2x80x83(1)
radius={square root over ( )}(g2+h2xe2x88x92c)xe2x80x83xe2x80x83(2)
In the three dimensional measurement, the surface roughness (RA) and the minimum-maximum value (P-V value) are additionally measured.
Conventionally, in the three dimensional measurement, there are some measuring instruments for measuring the three dimensional wave front phase such as an interferometer which calculate and represent the wave front phase configuration as the amount of astigmatism. However, no measuring instrument has been proposed which measures the surface shape for a three dimensional curved surface shape such as a cylindrical or toroidal shape as the radiuses of curvature in the generating line and principal line direction and calculates the surface shape.
Thus, it is considered that the shape data of the three dimensional surface of the measurement object is acquired by two dimensional scanning, employing the conventional two dimensional measuring instrument. In this case, however, the curved surface shape that is not in rotation symmetry such as a cylindrical or toroidal shape is often unknown in the generating line direction or principal line direction. If the two dimensional scanning is not made along the generating line or principal line direction, an error occurs in calculating the radius of curvature that is caused by an angle error in its scanning direction, resulting in incorrect scanning and measurement.
In the light of the above-mentioned problems, it is an object of the present invention to provide a surface shape measuring apparatus and method in which the surface shape of a measured object having unknown principal line or generating line is approximated with a quadratic curve such as a circle, an ellipse, a hyperbolic function or a quadratic function, and the surface shape of the measurement object is represented as a numerical value, on the basis of the center coordinates and the radius of curvature for the approximated quadratic curve.
In order to accomplish the above object, according to one aspect of the present invention, there is provided a surface shape measuring apparatus, characterized by polar coordinate conversion means for converting the surface shape data of a measurement object represented as the three dimensional orthogonal coordinate data (x, y, z) into the polar coordinate data (z, xcfx81, xcex8); approximate expression calculating means for calculating an approximate expression for the surface shape of the measurement object on the basis of the polar coordinate data, employing a polynomial for approximating the curved surface; angle calculating means for calculating an angle xcex81 representing the generating line direction and an angle xcex82 representing the principal line direction in the surface shape of the measurement object in accordance with the approximate expression; first approximate sectional shape calculating means for calculating first approximate sectional shapes both in the generating line and principal line directions on the surface shape of the measurement object on the basis of the angles xcex81 and xcex82 employing the approximate expression; second approximate sectional shapes calculating means for approximating the first approximate sectional shapes to the quadratic curves with respect to the three dimensional orthogonal coordinates as second approximate sectional shapes, and calculating the center coordinates and the radiuses of curvature for the approximate quadratic curves representing the second approximate sectional shapes in the generating line and principal line directions; and sectional shape data calculating means for calculating numerical data of the second approximate sectional shapes both in the generating line and principal line directions and on the surface shape of the measurement object on the basis of the center coordinates and the radiuses of curvature for the approximate quadratic curves.
The approximate expression may be a Zernike""s polynomial including at least up to nine terms.
According to another aspect of the invention, there is provided A surface shape measuring method comprising the steps of: converting the surface shape data of a measurement object represented as the three dimensional orthogonal coordinate data (x, y, z) into the polar coordinate data (z, xcfx81, xcex8); calculating an approximate expression for the surface shape of the measurement object on the basis of the polar coordinate data employing a polynomial for approximating the curved surface; calculating an angle xcex81 representing the generating line direction and an angle xcex82 representing the principal line direction in the surface shape of the measurement object in accordance with the approximate expression; calculating first approximate sectional shapes both in the generating line and principal line directions on the surface shape of the measurement object on the basis of the calculated angles xcex81 and xcex82, employing the approximate expression; approximating the first approximate sectional shapes both in the generating line and principal line directions to quadratic curves with respect to the three dimensional orthogonal coordinates as second approximate sectional shapes and calculating respective center coordinates and the radiuses of curvature for the approximate quadratic curves; and calculating numerical data of the second approximate sectional shapes both in the generating line and principal line directions in the surface shape of the measurement object, on the basis of the center coordinates and the radiuses of curvature for the approximate quadratic curves.
The approximate expression may be a Zernike""s polynomial including at least up to nine terms.